Ratio and Proportion

Ratio is a comparison of two quantities by division. It tells us how much of one thing there is compared to another. For example, if a class has 20 boys and 30 girls, the ratio of boys to girls is 20:30, which can be simplified to 2:3.

A proportion is an equation stating that two ratios are equal. For instance, if 2 out of 5 students are girls, and we want to know how many girls there are in a group of 20 students, we can set up a proportion: 2/5 = x/20.

Types of Ratios

When dealing with ratios, we often encounter different types:

• Ratio of Equality: Both quantities are equal (e.g., 5:5 or 1:1).
• Ratio of Greater Inequality: The first quantity is greater than the second (e.g., 5:2).
• Ratio of Less Inequality: The first quantity is less than the second (e.g., 2:5).

Continued Proportion

Three quantities a, b, and c are said to be in continued proportion if a:b = b:c. This means b is the mean proportional between a and c. Mathematically, b² = ac.

Important Concepts and Formulas

Let two numbers be 'a' and 'b'.

• Ratio of a to b: a : b or a/b
• Inverse Ratio of a : b: b : a or b/a
• Compound Ratio: The ratio of the products of the antecedents to the products of the consequents of two or more ratios. For example, the compound ratio of a:b and c:d is ac:bd.
• Proportion: If a:b = c:d, then a, b, c, d are in proportion. This can be written as a/b = c/d. Here, 'a' and 'd' are called extremes, and 'b' and 'c' are called means. The product of extremes equals the product of means: a × d = b × c.

Examples of Ratio and Proportion Problems

Example 1: The ratio of two numbers is 3:7. If the sum of the numbers is 100, find the numbers.
Let the numbers be 3x and 7x.
3x + 7x = 100
10x = 100
x = 10
The numbers are 3 * 10 = 30 and 7 * 10 = 70.

Example 2: Divide $1200 among A, B, and C such that A:B = 2:3 and B:C = 4:5.
To combine these ratios, we need to make the 'B' part common.
A:B = 2:3 = (2*4):(3*4) = 8:12
B:C = 4:5 = (4*3):(5*3) = 12:15
So, A:B:C = 8:12:15.
The total parts are 8 + 12 + 15 = 35.
A's share = (8/35) * 1200
B's share = (12/35) * 1200
C's share = (15/35) * 1200

Simple Interest

Simple Interest (SI) is a method of calculating the interest charged on a loan or earned on an investment. It is calculated on the principal amount only. The interest amount remains the same for each year of the loan or investment period.

Key Terms

• Principal (P): The initial amount of money borrowed or invested.
• Rate of Interest (R): The percentage at which interest is charged or earned, usually per annum.
• Time (T): The duration for which the money is borrowed or invested, usually in years.
• Simple Interest (SI): The interest earned or paid on the principal amount.
• Amount (A): The total sum of money after adding the simple interest to the principal (A = P + SI).

Formula for Simple Interest

The formula to calculate Simple Interest is:
SI = (P × R × T) / 100

Where:
P = Principal amount
R = Rate of interest per annum (in %)
T = Time period (in years)

From this, we can derive formulas to find P, R, or T if the other values are known:
P = (SI × 100) / (R × T)
R = (SI × 100) / (P × T)
T = (SI × 100) / (P × R)

The Amount (A) after T years is:
A = P + SI
A = P + (P × R × T) / 100
A = P (1 + RT/100)

Examples of Simple Interest Problems

Example 1: Calculate the Simple Interest on $5000 at a rate of 8% per annum for 3 years.
P = $5000
R = 8%
T = 3 years
SI = (5000 × 8 × 3) / 100
SI = (50 × 8 × 3)
SI = 400 × 3
SI = $1200
Amount (A) = P + SI = 5000 + 1200 = $6200

Example 2: In what time will $2000 amount to $2600 at 5% simple interest per annum?
P = $2000
A = $2600
SI = A - P = 2600 - 2000 = $600
R = 5%
Using the formula T = (SI × 100) / (P × R):
T = (600 × 100) / (2000 × 5)
T = (60000) / (10000)
T = 6 years

Compound Interest

Compound Interest (CI) is the interest calculated on the initial principal amount and also on the accumulated interest from previous periods. In essence, it's "interest on interest." This means the principal amount grows over time, leading to a higher total return compared to simple interest over the same period.

Key Terms

• Principal (P): The initial amount of money borrowed or invested.
• Rate of Interest (R): The percentage at which interest is compounded, usually per annum.
• Time (T): The duration for which the money is borrowed or invested, usually in years.
• Compounding Period: The frequency at which interest is added to the principal (e.g., annually, semi-annually, quarterly, monthly).
• Amount (A): The total sum of money after adding the compound interest to the principal.
• Compound Interest (CI): The total interest earned over the period. CI = A - P.

Formula for Compound Interest

When interest is compounded annually:
Amount (A) = P (1 + R/100)^T
Compound Interest (CI) = A - P
CI = P (1 + R/100)^T - P
CI = P [(1 + R/100)^T - 1]

Compounding Frequency

If the interest is compounded more frequently than annually, we adjust the rate and time:

• Semi-annually (Half-yearly): The rate is halved (R/2), and the time is doubled (2T).
  A = P (1 + (R/2)/100)^2T
• Quarterly: The rate is divided by four (R/4), and the time is multiplied by four (4T).
  A = P (1 + (R/4)/100)^4T
• Monthly: The rate is divided by twelve (R/12), and the time is multiplied by twelve (12T).
  A = P (1 + (R/12)/100)^12T

In general, if interest is compounded 'n' times per year:
A = P (1 + R/(100n))^nT

Examples of Compound Interest Problems

Example 1: Find the compound interest on $10,000 for 2 years at 5% per annum, compounded annually.
P = $10,000
R = 5%
T = 2 years
A = P (1 + R/100)^T
A = 10000 (1 + 5/100)²
A = 10000 (1 + 0.05)²
A = 10000 (1.05)²
A = 10000 × 1.1025
A = $11,025
CI = A - P = 11025 - 10000 = $1025

Example 2: Calculate the amount and compound interest on $8,000 for 1.5 years at 10% per annum, compounded semi-annually.
P = $8,000
R = 10% per annum
T = 1.5 years
Compounded semi-annually means n = 2.
The rate per period = R/n = 10%/2 = 5% = 0.05
The number of periods = nT = 2 × 1.5 = 3
A = P (1 + (R/n)/100)^nT or A = P (1 + Rate per period)^{Number of periods}
A = 8000 (1 + 5/100)³
A = 8000 (1 + 0.05)³
A = 8000 (1.05)³
A = 8000 × 1.157625
A = $9261
CI = A - P = 9261 - 8000 = $1261

Area and Volume

Area refers to the amount of two-dimensional space a shape occupies, measured in square units (e.g., cm², m², sq ft). Volume refers to the amount of three-dimensional space an object occupies, measured in cubic units (e.g., cm³, m³, cubic ft).

Common Geometric Shapes and Formulas

2D Shapes (Area)

Shape	Formula for Area	Variables
Square	Area = side2	side (s)
Rectangle	Area = length × width	length (l), width (w)
Triangle	Area = ½ × base × height	base (b), height (h)
Circle	Area = πr2	radius (r), π ≈ 22/7 or 3.14
Parallelogram	Area = base × height	base (b), height (h)
Trapezium	Area = ½ × (sum of parallel sides) × height	parallel sides (a, b), height (h)

3D Shapes (Volume and Surface Area**)**

**Note:** Surface Area is the total area of the outer surfaces of a 3D object.

Shape	Formula for Volume	Formula for Surface Area (Total)	Variables
Cube	Volume = side3	Surface Area = 6 × side2	side (s)
Cuboid (Rectangular Prism)	Volume = length × width × height	Surface Area = 2(lw + lh + wh)	length (l), width (w), height (h)
Sphere	Volume = (4/3)πr3	Surface Area = 4πr2	radius (r), π ≈ 22/7 or 3.14
Cylinder	Volume = πr2h	Surface Area = 2πr(r + h)	radius (r), height (h), π ≈ 22/7 or 3.14
Cone	Volume = (1/3)πr2h	Surface Area = πr(r + l) where l = √(r2 + h2) (l is slant height)	radius (r), height (h), slant height (l), π ≈ 22/7 or 3.14

Examples of Area and Volume Problems

Example 1: Find the area of a rectangular garden that is 15 meters long and 8 meters wide.
Area = length × width
Area = 15 m × 8 m
Area = 120 m²

Example 2: A cylindrical tank has a radius of 7 meters and a height of 10 meters. Calculate its volume. (Use π = 22/7)
Volume = πr²h
Volume = (22/7) × (7 m)² × 10 m
Volume = (22/7) × 49 m² × 10 m
Volume = 22 × 7 m² × 10 m
Volume = 154 m² × 10 m
Volume = 1540 m³

Example 3: What is the cost of painting the four walls of a room 5m long, 4m wide and 3m high at $10 per square meter?
This involves the lateral surface area of a cuboid (area of the 4 walls).
Lateral Surface Area = 2(l + w)h
Lateral Surface Area = 2(5m + 4m) × 3m
Lateral Surface Area = 2(9m) × 3m
Lateral Surface Area = 18m × 3m
Lateral Surface Area = 54 m²
Cost of painting = Area × Rate per square meter
Cost = 54 m² × $10/m²
Cost = $540

Time and Work

The Time and Work topic deals with problems where individuals or groups complete a task. It is based on the concept that more workers mean less time, and fewer workers mean more time, assuming they all work at the same rate.

Key Concepts

• Work Done: The amount of task completed.
• Rate of Work: The amount of work done per unit of time. If a person can complete a work in 'x' days, their rate of work is 1/x of the work per day.
• Total Work: Usually considered as 1 unit (representing the whole task).

Formulas and Relationships

1. If a person can do a piece of work in 'x' days, then the work done by them in one day is 1/x.
Example: If Ram can complete a work in 10 days, he does 1/10 of the work each day.

2. If a person does 1/y of a work in one day, then they will take 'y' days to complete the whole work.
Example: If Sita does 1/5 of the work each day, she will take 5 days to complete the work.

3. When multiple people work together: If A can do a work in x days and B can do the same work in y days, then working together, the time taken is:
Time together = 1 / (Rate of A + Rate of B)
Time together = 1 / (1/x + 1/y)
Time together = xy / (x + y) days

4. When people leave or join: Problems might involve individuals working for some days and then leaving, or new people joining. In such cases, calculate the work done by each person for the days they worked and sum it up.

Examples of Time and Work Problems

Example 1: A can do a piece of work in 10 days and B can do the same work in 15 days. In how many days can they together finish the work?
Time taken by A = 10 days. Work done by A in 1 day = 1/10.
Time taken by B = 15 days. Work done by B in 1 day = 1/15.
Work done by A and B together in 1 day = 1/10 + 1/15
= (3 + 2) / 30 = 5/30 = 1/6
Time taken by them together to finish the work = 1 / (1/6) = 6 days.
Using shortcut: xy / (x+y) = (10 * 15) / (10 + 15) = 150 / 25 = 6 days.

Example 2: A can complete a work in 18 days, B in 24 days, and C in 30 days. They work together for 5 days. After that, B leaves. In how many days will A and C complete the remaining work?
Work done by A in 1 day = 1/18
Work done by B in 1 day = 1/24
Work done by C in 1 day = 1/30
Work done by A, B, C together in 1 day = 1/18 + 1/24 + 1/30
LCM of 18, 24, 30 is 360.
= (20 + 15 + 12) / 360 = 47/360
Work done by A, B, C in 5 days = 5 × (47/360) = 235/360 = 47/72.
Remaining work = 1 - 47/72 = (72 - 47) / 72 = 25/72.
Now, A and C work together.
Work done by A and C in 1 day = 1/18 + 1/30
LCM of 18, 30 is 90.
= (5 + 3) / 90 = 8/90 = 4/45.
Time taken by A and C to complete the remaining work (25/72) = Remaining Work / (Work done by A & C per day)
= (25/72) / (4/45)
= (25/72) × (45/4)
= (25 × 45) / (72 × 4)
= (25 × 5) / (8 × 4) (dividing 45 and 72 by 9)
= 125 / 32 days.